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Spontaneous Localization of Vibrational Energy D Spontaneous Localization of Vibrational Energy D. Brown, L. Bernstein To cite this version: D. Brown, L. Bernstein. Spontaneous Localization of Vibrational Energy. Journal de Physique IV Proceedings, EDP Sciences, 1995, 05 (C4), pp.C4-461-C4-474. 10.1051/jp4:1995437. jpa-00253742 HAL Id: jpa-00253742 https://hal.archives-ouvertes.fr/jpa-00253742 Submitted on 1 Jan 1995 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE IV Colloque C4, supplkment au Journal de Physique 111, Volume 5, mai 1995 Spontaneous Localization of Vibrational Energy D.W. Brown and L. Bernstein* Institute for Nonlinear Science, University of California, Sun Diego La Jolla, CA 92093-0402, U.S.A. * Department of Mathematics, Box 8085, Idaho State University, Pocatello, ID 83209-8085, U.S.A. ABSTRACT --- In this paper we illustrate a number of the characteristic manifestations of soft anharmonicity in the behavior of lattice vibrations, both in and out of thermal equilibrium. In particular, we focus on various ways in which vibrational energy may come to be localized as a consequence of anharmonicity, even in defect-free lattices. These include the "overpopulation" of anharmonic vibrations in thermal equilibrium, the inhibition of dispersion, and the enhancement of spatial coherence. The relevance of the spontaneous localization of vibrational energy in the formation of "hot spots" is discussed, with particular emphasis on the unstable evolution of high-amplitude initial conditions as may be found in initiating shocks. Research is what l'm doing when I don't know what l'm doing. -- Werner von Braun 1. INTRODUCTION The process of detonation is remarkable in part because of its strong linkage of microscopic events and macroscopic dynarnical effects. This cascade across space and time scales has as one of its consequences that no one scientific discipline embraces the phenomenon in its entirety. This paper is addressed to the anharmonic behavior of crystal vibrations in the initiation phase of detonation, defined as the interval between the delivery of a low-amplitude shock into virgin material and the later development of a steady-state detonation front. The phenomenon of greatest interest here is the localization of vibrational energy, particularly whether such localization may facilitate the initiation of chemistry that may ultirnatety support detonation. While it is possible that such localization may also be involved in the steady-state propagation of the detonation front, the strong role of chemistry in the steady-state problem suggests at least that the space-time interval in which anharmonic vibrational effects may be distinguishable as such may be strongly reduced relative to the initiation phase. In a general sense, the subject to be developed here may be associated with the established concept of "hot spots"; however, it is important to emphasize an essential distinction between the usual conception of hot spots and that of anharmonic localization. While many implementations of the idea could be enumerated, the established concept of hot spots is almost universally associated with material defects that become superheated when stressed by an impinging shock. Though explosive sensitivity is strongly influenced by inhomogeneities of diverse origins, crystalline or nearly defect-free energetic materials are also found to be detonable, albeit at elevated shock pressures. This suggests that even in homogeneous materials processes exist that may be capable of achieving significant energy localization under the influence of a sufficiently energetic drive. This is precisely the nature of soft anharmonic vibrations. A vibration may be characterized as soft (hard) if the vibrational frequency decreases (increases) with increasing amplitude. In quantum-mechanical terns, a soft (hard) vibration is one whose energy level spacing decreases (increases) with increasing vibrational quantum number. These characterizations apply Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1995437 C4-462 JOURNAL DE PHYSIQUE IV equally to symmetric potentials as we use in this paper, or to asymmetric potentials such the Morse or Lennard-Jones [I]. In the case of vibrations whose potentials are symmetric with respect to the equilibrium amplitude, a soft (hard) vibration is one whose potential increases more slowly (rapidly) with increasing amplitude than does the harmonic reference potential describing small oscillations in the same oscillator. These characterizations may change as a function of amplitude; consider, for example, an oscillator whose potential first increases more slowly than harmonic, but which later increases more rapidly than harmonic. For the smooth potentials characteristic of mechanical equilibria, such changes the character of the anharmonicity occur only above some kite amplitude, such that a vibration may be characterized uniquely as soft or hard depending on the leading anharmonic correction at low amplitudes; it is this characterization that we apply throughout this paper. 2. THERMODYNAMICS It will be useful in the following to consider the anharmonic oscillator described by the Hamiltonian for soft anharmonicity, E > 0. (It is well to note that this oscillator is thermodynamically unstable, since the potential is unbounded below as Ix I increases beyond x, = E-", and should be handled with care when used in more than a qualitative sense.) The equipartition theorem in its general form [2] reads Using the anharmonic oscillator Hamiltonian (I), this yields which indicates that the equilibrium expectation value of the total energy of a soft anharmonic oscillator exceeds that of a harmonic oscillator. This qualitative conclusion does not depend on the frequency of the harmonic oscillator used for comparison, nor on the degree of freedom underlying the vibration, since all harmonic oscillators in equilibrium at the same temperature share the same energy expectation value. Consequently, we may conclude that any soft anharmonic oscillator will equilibrate to an average energy higher than that of any harmonic oscillator at the same temperature. The magnitude of this excess is not necessarily large, of course. Clearly, the magnitude of this disparity increases with the strength of the anharmonicity, but for the same value of the anharmonicity (e.g., the same E), the expected value of the higher moments (e-g. ar4>)will be larger in oscillators having lower frequencies at Iow amplitude (see below). This suggests that we may expect anharmonicity-driven con- centrations of energy to be most pronounced among the lowest-lying vibrational modes of a complex sys- tem. It is well to note that because the principal effect of an impinging shock is to promptly compress material passing under it, energy is first delivered into low-lying vibrational modes associated with volumetric compression; these include the acoustic modes associated with translations of the center of mass of a unit cell, librational modes associated with rigid-body rotations of principal unit cell constituents, and the lowest-lying optical modes associated with relative translations of unit cell constituents. The remaining modes of vibration are associated more essentially with deformations of individual molecules, and being generally substantially higher in frequency, couple less effectively with shocks. (For cautionary observa- tions in this regard, see e.g. Ref. [3].) Sufficiently far behind the shock, we may assume the material to come to equilibrium at a temperature substantially higher than that of the unshocked material. Under such conditions, the intrinsic anharmoni- city of any vibration should be more pronounced, with the strongest anharmonic effects evident in the low-lying vibrations. In the following, we make use of the notion of a "reference oscillator", by which we mean a hypothetical harmonic oscillator in the same degree of freedom as the anhannonic oscillator in question, such that the frequency of this harmonic oscillator matches the frequency of the anharmonic oscillator at small amplitudes. This allows us to focus on the manner in which the properties of anharmonic oscillators deviate from the harmonic ideal. In such comparisons, quantities associated with anhannonic oscillators are decorated with tildes (e.g., V),while harmonic reference quantities are decorated with zeros (e.g., Vo) to indicate that the anharmonicity has been set to zero. Moreover, for simplicity we explicitly consider only vibrational potentials symmetric about the equilibrium. Since the potential of a soft anharmonic oscillator is less than or equal to that of the harmonic reference oscillator at all amplitudes (v(x) 4 Vo(x)), it is evident that the partition function Q of an anharmonic oscillator should be greater than the partition function Qo of the harmonic reference oscillator at all temperatures; i.e., = e-[y~'+Y(x)~~kB~ -1'4'+ V,(x)]lk,T II dpdx = Q,. Consequently, the free energy of the soft anharmonic oscillator is lower (4 <Ao), and (using the equipartition
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